A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

PLANE LOCI 189
inflected lies on a circle given in position.' The meaningseems
to be this : Given two fixed points A, B a length a,
y
a straight line OX with a point fixed upon it, and a direction
represented, say, by any straight line OZ through 0, then,
if AP, BP be drawn to P, and PM parallel to OZ meets OX
in M, the locus of P will be a circle given in position if
a.AP 2 + /3.BP = 2 a.0M,
where a, /3 are constants. The last two loci are again
obscurely expressed, but the sense is this :
(7) If PQ be any
chord of a circle passing through a fixed internal point 0, and
R be an external point on PQ produced such that either
(a) OR 2 = PR.RQ or (b) 0R 2 + P0 . 0Q= PR . RQ, the locus
of R is a straight line given in position. (8) is the reciprocal
of this : Given the fixed point 0, the straight line which is
the locus of R, and also the relation (a) or (b), the locus of
P, Q is a circle.
(£) Nevcreis (Vergings or Inclinations), two Books.
As we have seen, the problem in a vevo-is is to place
between two straight lines, a straight line and a curve, or
two curves, a straight line of given length in such a way
that it verges towards a fixed point, i.e. it will, if produced,
pass through a fixed point. Pappus observes that,
when we come to particular cases, the problem will be
' ', plane ',
'
solid or linear ' according to the nature of the
'
particular hypotheses ; but a selection had been made from
the class which could be solved by plane methods, i.e. by
means of the straight line and circle, the object being to give
those which were more generally useful in geometry. The
following were the cases thus selected and proved. 1
I. Given (a) a semicircle and a straight line at right angles
to the base, or (b) two semicircles with their bases in a straight
line, to insert a straight line of given length verging to an
angle of the semicircle [or of one of the semicircles].
II. Given a rhombus with one side produced, to insert
a straight line of given length in the external angle so that it
verges to the opposite angle.
1
Pappus, vii, pp. 670-2.
190 APOLLONIUS OF PERGA
III.
Given a circle, to insert a chord of given length verging
to a given point.
In Book I of Apollonius's work there were four cases of
I (a), two cases of III, and two of II ; the second Book contained
ten cases of I (b).
Restorations were attempted by Marino Ghetaldi (Apollonius
redivivus, Venice, 1607, and Apollonius redivivus . . . Liber
secundus, Venice, 1613), Alexander Anderson (in a Suppleonentum
Apollonii redivivi, 1612), and Samuel Horsley
(Oxford, 1770); the last is much the most complete.
In the case of the rhombus (II) the construction of Apollonius
can be restored with certainty. It depends on a lemma given
by Pappus, which is as follows : Given a rhombus AD with
diagonal BG produced to E, if F be taken on BG such that EF
is a mean proportional between BE and EG, and if a circle be
described with E as centre and EF as radius cutting CD
in K and AG produced in H, then shall B, K, H be in one
straight line. 1
Let the circle
join HE, LE, KE.
cut AG in L, join LK meeting BG in M, and
Since now CL, GK are equally inclined to the diameter of
the circle, CL = GK. Also EL = EK, and it follows that the
triangles ECK, ECL are equal in all respects, so that
or
By hypothesis,
,
Z CKE = L CLE = Z CHE.
EB:EF=EF: EC,
EB:EK = EK:EC.
1
Pappus, vii, pp. 778-80.